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73
THE FRECHET DIFFERENTIABILITY
OF CONVEX FUNCTIONS ON C(S)
Roger Eyland and Bernice Sbarp
INTRODUCTION
In this paper we continue our study of the differentiability of convex functions with
domain in a topological linear space: here we are particularly concerned with Frechet
differentiability and the space C(S), where S is an arbitrary topological space. The
maximum functions, defined by mA = suptEA x(t) (A C S compact), are good test
functions: we show that the Frechet differentiability points of mA are precisely those
functions which attain their maximum on A at a single isolated point of A; mA is Fn\chet
differentiable on a dense subset of C( S) if and only if the set of isolated points of A is
dense in A. Relationships between the seminorms PA and the corresponding maximum
functions are given. The set of Frechet differentiability points of mA, and hence of PA,
is open; this generalises the well known result that for compact S the sup norm on C( S)
is Frechet differentiable on an open set. In a topological section, connections between
the structure of thin and dispersed sets lead into an exploration of the topology of S
for Asplund C(S) spaces. In particular, we prove that C(Q) is an Asplund space.
1991 Mathematics Subject Classification. 58C20, 46G05.
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Background
For S compact, C(S) is a Banach space and the sup norm topology coincides with the -
topology of compact convergence. Cohan and Kenderov study the Gateaux differentia
bility of the maximum functions mA on the Banach space C( S) and give a condition on
S for the set of points of Gateaux differentiability of the norm to be dense, but contain
no G6 subset [1,3.3]. Namioka and Phelps prove that for compactS, C(S) is Asplund
if and only if S is dispersed [7, 18]. This result is generalised to C(S) for completely
regular S in [3, 3.8]: C(S) is Asplund if and only if every compact subset of S is dis
persed. It is a consequence of a generalisation of Mazur's Theorem [10, 2.1] that C(S)
is Weak Asplund whenever Sis a-compact and metrisable, for example C(R) is Weak
Asplund.
Our first results are about the Frechet differentiability of the maximum functions: it
is easy to show that mA is Gateaux differentiable at x E C(S) if and only if x attains its
maximum on A at only one point; the equivalent condition for Frechet differentiability is
that x attains a maximum at one isolated point of A. We then illustrate how results for a
defining family of seminorms can be retrieved from the maximum functions. Properties
of thin and dispersed sets are explored in a topological interlude which highlights the
structure of S for Asplund C(S) spaces. In Section 4 direct proofs are given of some of
the theorems of [3], which are there deduced from Banach space theory. Section 5 is
devoted to examples; techniques of [10] and [3] are used and in particular we exhibit
our favourite Asplund space, C(Q).
Preliminaries
Spaces are assumed Hausdorff; the term function is used for a real valued map.
Let U be an open subset of a topological linear space X and let M be a homology on
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X, that is, a class of bounded subsets containing all singletons. A real valued function
f on U is M -differentiable at x E U whenever there exists u E X* such that, for all
ME M, for all € > 0, there exists 8 > 0, such that for all y E M, for all t : ltl E (0, 8),
I c::......:f ('-x _+ --=-ty.:_) -___:!:....::.( x--'-) ( ) I - t - u y < €.
The map u is uniquely determined by f and x and is denoted by f'(x).
If M is the class of all bounded (singleton) subsets of X then f is Frechet (Gateaux)
differentiable at x.
A function f defined an open convex subset D of a topological linear space X is
said to be convex whenever, for all x, y E D, for all t E [0, 1],
f(tx + (1- t)y)::; tf(x) + (1- t)f(y).
A continuous convex function f is Gateaux differentiable at x E D if and only if for all
y EX,
¢(y) =lim f(x + ty)- f(x) t-o t
exists; the continuity and linearity of ¢ are consequences of the continuity and convexity
of f.
For Sa topological space, C(S) denotes the set of all real valued continuous func:
tions on S with the topology of compact convergence, that is, the topology generated
by the seminorms
PA(x) = max{lx(t)!: tEA},
where A ranges over all nonempty compact subsets of S.
The maximum function mA: C(S)---. R, defined for each nonempty compact subset
A of S by
mA(x) = max{x(t): tEA},
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is continuous and convex on C(S).
(There is no loss of generality in assuming S to be completely regular: for any
topological space X there is a completely regularS such that C(X) is linearly isomorphic
to C(S) [4,3.9].)
1. FRECHET DIFFERENTIABILITY OF THE MAXIMUM FUNCTIONS
In this section we see that Fn§chet differentiability of the maximum functions is rare. A
function x E C(S) is a Frechet differentiability point of mA if and only if x attains its
maximum on A at precisely one isolated point of A; mA is Frechet differentiable on a
dense subset of C( S) if and only if the set of isolated points of A is dense in A.
For compact S, 1.1 is [1, Proposition 1.2]. No proof is given here since extending to
completely regular S requires only trivial changes: for the if case the complete regularity
of S ensures the existence of a suitable Urysohn function in C(S) and for the only if
even this is superfluous.
1.1. The maximum function mA is Gateaux differentiable at x E C(S) if and only if x
attains its maximum at only one point of A. If x attains its maximum only at t 0 E A,
the derivative of m A at x is the evaluation function 8t 0 : C( S) --Jo R defined by
The corresponding condition for Frechet differentiability is that this maximum oc
curs at an isolated point of A.
1.2. Suppose S is a completely regular space and A is a nonempty compact subset
of S. The function mA is Frechet differentiable at x 0 E C(S) if and only if x 0 has a
maximum on A at precisely one isolated point of A.
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Proof. Assume that x0 E C(S) attains its maximum on A only at the isolated point
t0 • Then A\ {to} is closed and so compact, hence x0 attains its maximum on A\ {to}
at some point t 1 E A\ {to}, and xo(ti) < xo(to). Let k = xo(to)- xo(ti).
Let M be a bounded subset of C(S). There exists c > 0 such that for ally EM
and for all tEA, Jy(t)J <c. Let 8 = tc· For ally EM, for all A E (0,8), for all tEA,
l(xo + Ay)(t)- xo(t)i = Ajy(t)i
and specifically,
(xo + Ay)(to) > xo(to)- tk.
For all tEA\ {to}, for ally EM, from(*),
(xo + Ay)(t) < xo(t) + tk
= xo(to)- tk, (where the last line follows from the definition of k) which with ( **) implies that for all
tEA\ {to}
(xo + Ay)(t) < (xo + Ay)(to).
Hence (x0 + Ay) attains a maximum on A only at t 0 , and
mA(xo + AY) = (xo + Ay)(to),
so for ally EM, for all A E (0,8),
mA(xo + Ay)- mA(xo) A
(xo + Ay)(to)- xo(to) A
= y(to)
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and mA is Frechet differentiable at xo.
The plan of the proof of the converse goes like this. Assume that mA is Frechet
differentiable at x 0 ; then x0 has a maximum on A only at t0 . Assume that t 0 is not an
isolated point, the complete regularity of S, and the fact that to is not isolated, imply
the existence of a bounded sequence (Yn) in C(S) and a sequence (kn) of positive real
numbers converging to 0 such that
I mA(xo + knYn)- mA(xo) -li ( )I > 1 kn to Yn _ '
which implies that lito, the Gateaux derivative of mA at x 0 , is not the Frechet derivative.
Formally, since xo is continuous, for each n E N,
Un = x01 (xo(to) + (- 21n' 2~))
= {t E S: - 21n < Xo(to)- xo(t) < 21n}
is an open neighbourhood of t 0 •
Assume to is not an isolated point of A; for each n E N, there exists Sn E An Un
such that Sn #to and
-{n < xo(to)- xo(sn) < 21n,
but since xo has a maximum on A only at t0 and because sn # t 0 ,
0 < xo(to)- xo(sn).
Let
kn = 2(xo(to)- xo(sn))
so that 0 < kn <: ~· The sequence (kn) of positive real numbers converges to 0. Since
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so
Hence for each n E N, the set Dn defined by
is a.n open neighbourhood of Sn which does not contain to. By complete regularity, for
each n EN there exists Yn E C(S) such that
Yn(t) E [o, !] for all t E S,
Yn(t) = 0 for all t E S \ Dn
arid in particular Yn(to) = 0.
Since for any compact set [( C S and any n E N,
sup {Yn(t): t E K} :::; !,
(Yn) is a bounded sequence of points of C(S). Also
so
Since for all n EN, bt0 (Yn) = Yn(to) = 0,
mA(xo + knYn)- mA(xo) _ 8 ( ) > kn _ 0 k to Yn - k
n n
=1.
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Taking e = 1 and M = {yn} as the bounded set, for each li > 0 there exist kn E (0, li)
and Yn E M such that
lmA(xo + knYn)- mA(xo) -li ( )I>
kn to Yn _ e
so lito is not the Frechet derivative of mA. • A set is said to be perfect if it is closed and has no isolated points.
The following corollary is an immediate consequence of 1.2.
1.3 Corollary. Suppose S is completely regular and A a nonempty compact subset
of S; mA is nowhere Frechet differentiable if and only if A is perfect.
1.4. H S is completely regular, and A is a nonempty compact subset of S, the set of
points of Frechet differentiability of m A is an open subset of C( S).
Proof. Let A be a nonempty compact set inS, let D be the set of points of Frechet
differentiability of mA, and let xED. From 1.2, x has a maximum only at an isolated
point to E A. Let t 1 be any point where x attains a maximum on the compact set
A\ {to}.
Let e = x(to)- x(t1 ); then e > 0. Define V C C(S) by
V = {y E C(S): for all tEA, iy(t)i < !e};
then x +Vis a neighbourhood of x in C(S). If y Ex+ V, then
y(t0 ) > x(to)- !e
and for all tEA\ {to} y(t) < x(t) + !e
=x(t0 )-e+!e
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So y attains its maximum on A only at i 0 . From 1.2, mA is Frechet differentiable at y
so each point of x + V is in D. Since each point of D has a neighbourhood contained
in D, Dis open.
1.5. LetS be completely .regular and A a compact subset of S. The maximum function
rnA is Frechet differentiable on a dense subset of C(S) if and only if the set of isolated
points of A is dense in A.
Proof. Assume that mA is Frechet differentiable on a dense subset of C(S), let t 0 E A,
and let U be a neighbourhood of t0 in S. Then since S is completely regular, there
exists x E C( S) such that
x(t) E [0, 1] for aU i E S,
x(t0 ) = 1
x(t) = 0 for all t E S \ U.
Define V C C(S) by
V = {y E C(S): for all tEA, iy(t)i < t}.
The set x + V is a neighbourhood of x; by hypothesis there exists a Frechet differen-
tiability point z of rnA in x + V. From 1.2, z has a maximum on A only at an isolated
point, say t 1 . Now
and if t E ( S \ U) n A then
z(to) > x(t0 ) -- t = t
z(t) < x(t) + t 1
= 2·
It follows that z does not attain its maximum on A in An (S \ U), hence t1 is in U.
Thus each neighbourhood of t 0 contains an isolated point of A as required.
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Conversely, let A be a compact subset of S and let the set of isolated points of A
be dense in A. Let x E C( S) and let N be a neighbourhood of x in C( S). Then there
exist e > 0 and a compact subset K of S such that if
V = {y E C(S): for all t E K, iy(t)i < e}
then x + V C N. In particular, if hE C(S) satisfies the condition
for all t E S, ih(t)i < e,
then x+ hEN.
Assume that x attains a maximum on A at t0 ; then
is a neighbourhood of t 0 • By hypothesis, there exists an isolated point, t 1 , of A (which
could be t 0 ) contained in
that is
Since t1 is isolated in A, there is a neighbourhood U in S of t 1 which contains no other
point of UnA. By complete regularity there exists hE C(S) such that:
h(t) E [0, ¥1 for all t E S,
h(tl) = ¥, h(t) = 0 foralltES\U,
so in particular, x + h E N.
Since t1 is the only point of AinU, if tEA\ {tl},
(x + h)(t) = x(t) :5 mA(x).
However,
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(x + h)(ti) = x(t1 ) + h(ti)
>mA(x)-te+~e
= mA(x) + tE. Hence x + h has a maximum on A only at an isolated point t1 of A, from 1.2, mA is
Frechet differentiable at x+h, that is, at a point of N. Hence mA is Frechet differentiable
on a dense subset of C(S).
2. THE DIFFERENTIABILITY OF PA AND MA
Some of the differentiability properties of the family of seminorms which generate the
topology of compact convergence can be deduced from the differentiability properties
of the maximum functions. However, easy examples show that the differentiability of
either PA or mA at a point does not imply the differentiability of the other at that point.
2.1. Let S be a topological space, A a nonempty compact subset of S and M a
bornology on C ( S). The seminorm p A is J\.1 differentiable on a dense subset of C ( S) if
and only if the maximum function mA isM differentiable on a dense subset of C(S).
If mA isM differentiable on a dense open (dense G0 , open, G5) subset of C(S)
then so is PA·
Proof. If A is a singleton, say {t0 }, then the derivative of mA is 810 (see 1.1); mA
is M differentiable everywhere and PA is M differentiable everywhere except where
PA ( x) = 0. In this case all the theorem's conclusions hold. Assume that A has more
than one point.
Suppose that PAis M-differentiable on a dense subset of C(S). Let x E C(S), let
N be a neighbourhood of 0 in C(S) and let the minimum value of x on A. be k. There
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exist e > 0 and a compact set D containing A such that
B = {x E C(S): pv(x) < e} C N.
If U = x + e- k + B, there exists z E U such that PA is M-differentiable at z, but
on U, PA = mA, so mA is M-differentiable at z. Hence mA is M-differentiable at
z + k- E EX+ N.
To see the converse define D+,D- and Do by:
D+ = {x E C(S): mA(x) > mA(-x)}
D_ = {x E C(S): mA(x) < mA(-x)}
Do= {x E C(S): mA(x) = mA(-x)}
and let D = D+ U D_. Then, by continuity ofthe functions mA and x ~---+ mA( -x ), D+
and D_ are open in C(S), and D is dense in C(S).
If Gm is the set of M differentiability points of mA, then -Gm is the set of M
differentiability points of x ~--+.mA( -x). Let Gp be the set of M differentiability points
ofPA·
It is an easy consequence of the definitions that if two functions f and g agree on
an open set 0 then f is M differentiable at x E 0 if and only if g is. So,
and
By definition PA(x) = mA(!xl); if x E Do then mA(x) = mA(-x), and lxl has
at least two distinct points producing a maximum; from 1.1 x is not even a Gateaux
differentiability point of PA, that is Gp n Do is empty.
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Hence
If Gm is dense in C(S), then, since D+ is open,
and
So,
so,
that is, GP is dense in C(S).
The second half of the theorem follows from ( * ).
2.2. LetS be a compact space and C(S) the space of continuous functions on S with
sup norm topology. The set of points of Fnkhet differentiability of the sup norm is an
open set in C(S).
P:roof. From 1.4, taking A = S, the set of Frechet differentiability points of ms is
open; from 2.1 the set of FnSchet differentiability points of ps, which is the sup norm,
is open.
2.3. If Sis a compact space, the sup norm on C(S) is Fnkhet differentiable on a dense
set of C( S) if and only if the set of isolated points of S is dense in S.
P:roof. From 1.5 and 2.1, the following are equivalent:
(a) the set of isolated points of S is dense in S;
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(b) ms is Frechet differentiable on a dense set of C(S);
(c) ps, the sup norm, is Frechet differentiable on a dense set of C(S).
It was pointed out by Dr D. Yost that since the Stone-Cech compactification of the
integers is the closure of its isolated points, as a special case of 2.2 and 2.3 we have the
well known result that the natural norm on £00 is Frechet differentiable on a dense open
set.
3. THIN AND DISPERSED SETS
In this section we look at features of the topology of S which will give good Fn~chet
differentiability properties for mA on C( S).
A subset A of a topological space is dispersed if every nonempty subset of A contains
a relatively isolated point and thin if the set of isolated points of A is dense in A (that
is, if A contains only relatively isolated points and cluster points of relatively isolated
points). In the literature dispersed is synonymous with scattered and clairseme ([6, Ch
1, section 9, VI]). Thin is specific to this paper.
3.1. If Sis a dispersed topological space, then all subsets of S are thin, a fortiori Sis
thin.
Proof. Let A be a nonempty subset of S, let x E A and let N be an open neighbourhood
of x. The set N n A is a nonempty subset of S so by hypothesis contains an isolated
point of N n A and so, since N is open, of A; hence x is either an isolated point of A or
a cluster point of isolated points of A.
The converse is not true in general.
3.2. Even when S is compact, S may be thin but not dispersed.
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Proof. Consider R2 with the usual topology. Define T by
T = {(pfq, 1/q): p,q E N,O < p ~ q, for p, q mutually prime}
U {(x, 0) : 0 ~ x ~ 1}.
Then T is a compact subset of R2 ; T is thin, since the points with nonzero second
coordinate are isolated points, and the rest are cluster points of isolated points. However
T is not dispersed, since
{(x,y): y = 0}
contains no isolated point. • 3.3. A topological spaceS is dispersed if every closed subset of Sis thin.
Proof. Let W be a nonempty subset of the topological space S; W is the disjoint
union of a dispersed set and a perfect set [6, Section 9, part IV, result 3], that is
W = D u P and D n P is empty
where Dis dispersed and Pis perfect. Since Pis closed, it is by hypothesis thin, which,
by the definition of perfect is impossible unless it is empty; hence W is dispersed and
nonempty, and therefore contains an isolated point. • Of course if S is compact, then in 3.3 "closed" can be equivalently replaced by
compact. From 3.3 and 3.1 we deduce 3.4.
3.4. A topological spaceS is dispersed if and only if every subset of Sis thin.
For a space to be dispersed, it is not sufficient that every compact subset be thin.
3.5. The space of rationals, Q, is not dispersed but every compact subset is dispersed,
and hence thin.
Proof. ~t is easy to see that Q is not dispersed, since Q itself contains no isolated
points. From 3.3 it suffices to show that every compact subset of Q consists only of
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isolated points and cluster points of isolated points. The following proof of this is due
to Emeritus Professor Edwin Hewitt.
Sl}ppose F is a compact subset of Q; then F is compact in R, so we may suppose
that F is an infinite countable compact subset of R. Then from [6] (see 3.3)
where Dis a dispersed set and P a perfect set. Any nonempty perfect set in R contains
a copy of Cantor's ternary set, with cardinality the continuum; but F is countable by
hypothesis, so cannot contain a nonempty perfect subset, that is, P is empty. Hence F
is dispersed; from 3.1, F is thin. • 3.6. H every compact subset of S is thin, then every compact subset of S is dispersed.
Proof. Suppose A is a compact subset of S and F is a closed subset of A; then F is
compact, so by hypothesis F is thin. Hence every closed subset of A is thin; from 3.3,
A is dispersed. •
4. C(S) AS A DIFFERENTIABILITY SPACE
A topological linear space X is said to be an Asplund space (a Frechet differentiability
space) whenever every continuous convex function with domain a nonempty open con
vex subset is Frechet differentiable on a dense G 6 (dense) subset of that domain. We
will use the abbreviations ASP and FDS. If in the above definitions "Gateaux" replaces
"Frechet", the spaces are known as Weak Asplund (WASP) and Gateaux Differentia
bility Space (GDS). If X is a "bound covering" space (for definition and properties see
[3, section 2 and 3.2]), for example a Banach space or C( S) for S completely regular,
then ASP and FDS coincide, because the set of points of Frechet differentiability of a
continuous convex function is then a G6 set.
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To begin our study of C(S) as a differentiability space, we have 4.1 and 4.2 imme
diately from 1.3 and 1.5 respectively.
4.1. If S is completely regular and contains a nonempty compact perfect set, then
C(S) is not FDS.
4.2. If S is completely regular and C(S) is FDS then every compact subset of S is
thin.
Namioka and Phelps proved that for S compact, the Banach space C(S) is ASP
if and only if S is dispersed [7, 18]. A generalisation of one direction of this theorem,
gained from 4.2 and 3.6, is given in 4.3; from which half of their result, 4.4, is an
immediate corollary.
4.3. If C( S) is ASP then every compact subset of S is dispersed.
4.4 Corollary. Suppose S is compact and C( S) llas the sup norm topology. If C( S)
is ASP tl1en S is dispersed.
Towards the converse of 4.3 we have 4.5.
4.5. If Sis a completely regular space and if every compact subset il of Sis thin (or,
equivalently, dispersed), then every PA is Frechet differentiable on a dense open subset
of C(S).
Proof. From 2.1 it suffices to prove the property for the maximum functions mA; the
result follows from 1.5.
It is tempting to hope that from the result that every p A is Frechet differentiable
on a dense open set, we can conclude that C(S) is ASP; however/(= is not even GDS,
yet its natural norm is Frechet differentiable on a dense open set (see the comment
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following 2.3). We do know that if each PA is Frechet differentiable everywhere, except
perhaps on ker p, then C(S) is ASP [3, 3.5].
The following corollary is now immediate.
4.6 Corollary. SupposeS is compact and C(S) bas tbe sup norm topology. If Sis
dispersed tben tbe sup norm on C(S) is Fnkhet differentiable on a dense open set.
Whilst a necessary condition for C(S) to be ASP is given in 4.2, (and it is in fact
sufficient), a full characterisation seems not to be available by these direct methods.
The full characterisation, proved by different methods in [3, 3.8], is stated in 5.3.
5. EXAMPLES
From 4.1 it is clear that C(R) is not FDS. From 5.2 we conclude that C(N) is ASP
which leaves the intermediate problem of the classification of C( Q), posed as an open
question in the second author's thesis, to complete the picture.
5.1 ([3, 3. 7]). Suppose Sis completely regular. Then C(S) is ASP if and only if for
all compact subsets A of S, C(A) is ASP.
Using Namioka and Phelps result [7, 18] and 3.6, we have 5.2
5.2 ([3, 3.7]). SupposeS is completely regular. Tben C(S) is ASP if and only if every
compact subset of Sis dispersed (or equivalently thin).
Thus if Sis a discrete space, then C( S) is ASP: in particular Rn = C( {1, 2, 3, ... , n})
and C(N) are ASP.
From 3.5 the compact subsets of Q are dispersed.
5.3. C(Q) is ASP.
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School of Mathematics and Statistics University of Sydney NSW 2006 AUSTRALIA
Australian Catholic University 40 Edward St North Sydney NSW 2060 AUSTRALIA